Mechanics of Material – II Lab · Figure 1-5: (a) 2-sided overhanging beam (b) 1-sided...

MECHANICS OF MATERIAL – II

Instructor Lab Manual

Lab In charge

Engr. Sheraz Ali

Graduate Assistant

Engr. Luqman Razzaq

Lab Attendant

M. Irfan

Department of Mechanical Engineering

University of Engineering and Technology

Lahore, KSK-Campus

2019

TABLE OF CONTENTS

TABLE OF CONTENTS ............................................................................................................ i

LIST OF FIGURES ............................................................................................................... viii

LIST OF TABLES ..................................................................................................................... x

1 Lab Session – 1 .................................................................................................................. 1

1.1 Objective ..................................................................................................................... 1

1.2 Apparatus .................................................................................................................... 1

1.3 Summary of Theory .................................................................................................... 1

1.4 Classification of Beams ............................................................................................... 1

1.4.1 Cantilever Beam................................................................................................... 2

1.4.2 Simply Supported Beam ...................................................................................... 2

1.4.3 Overhanging Beams ............................................................................................. 2

1.4.4 Propped Cantilever Beams ................................................................................... 3

1.4.5 Fixed Beams......................................................................................................... 3

1.4.6 Continuous Beam ................................................................................................. 4

1.5 Procedure ..................................................................................................................... 4

1.6 Observations and Calculations (Aluminum) ............................................................... 4

1.7 Graph ........................................................................................................................... 5

1.8 Observations and Calculations (Brass) ....................................................................... 5

1.8.1 Graph.................................................................................................................... 6

1.9 Industrial Applications ................................................................................................ 6

1.10 Statistical Analysis .................................................................................................. 7

1.11 Conclusion ............................................................................................................... 7

1.12 Comments ................................................................................................................ 7

2 Lab Session – 2 .................................................................................................................. 8

2.1 Objective ..................................................................................................................... 8

ii

2.2 Apparatus .................................................................................................................... 8

2.3 Theory ......................................................................................................................... 8

2.3.1 Necking ................................................................................................................ 9

2.3.2 Bending .............................................................................................................. 10

2.3.3 Torsion ............................................................................................................... 11

2.3.4 Combine effect of bending and torsion .............................................................. 11

2.4 Bending ..................................................................................................................... 12

2.4.1 Procedure ........................................................................................................... 12

2.4.2 Observations and Calculations ........................................................................... 12

2.4.3 Statistical Analysis ............................................................................................. 13

2.5 Torsion ...................................................................................................................... 14

2.5.1 Procedure ........................................................................................................... 14



2.6 Combine Effect of bending and torsion .................................................................... 15

2.6.1 Procedure ........................................................................................................... 15



2.7 Industrial Applications .............................................................................................. 17

2.8 Comments.................................................................................................................. 17

2.9 Conclusion ................................................................................................................. 17

3 Lab Session – 3 ................................................................................................................ 18

3.1 Objective ................................................................................................................... 18

3.2 Apparatus .................................................................................................................. 18

3.3 Summary of Theory: ................................................................................................. 18

3.3.1 Curved bars / deflection of curved bars ............................................................. 18

3.3.2 Castiglione’s Theorem ....................................................................................... 19

iii

3.3.3 Derivation of the Castigliano’s Theorem for Quarter Circular Bar ................... 21

3.4 Procedure:.................................................................................................................. 24

3.5 Observations & Calculations: ................................................................................... 24

3.5.1 Specimen calculations ........................................................................................ 24

3.5.2 Graph.................................................................................................................. 24

3.6 Statistical Analysis .................................................................................................... 26

3.7 Industrial Applications: ............................................................................................. 26

3.8 Comments.................................................................................................................. 26

4 Lab Session – 4 ................................................................................................................ 27

4.1 Objective ................................................................................................................... 27

4.2 Apparatus .................................................................................................................. 27

4.2.1 Curved Bars ....................................................................................................... 27

4.3 Derivation of formulae .............................................................................................. 28

4.3.1 Derivation of the Castigliano’s Theorem for Semi-Circular Bar ....................... 28

4.4 Procedure ................................................................................................................... 31

4.5 Observations & Calculations ..................................................................................... 31

4.5.1 Specimen calculations ........................................................................................ 31

4.5.2 Graph.................................................................................................................. 32

4.6 Statistical Analysis .................................................................................................... 33

4.7 Industrial Applications .............................................................................................. 33

4.8 Comments.................................................................................................................. 33

5 Lab Session – 5 ................................................................................................................ 35

5.1 Objective ................................................................................................................... 35

5.2 Apparatus .................................................................................................................. 35

5.3 Theory ....................................................................................................................... 35

5.3.1 Hardness Tests ................................................................................................... 35

5.3.2 Rockwell Hardness Test .................................................................................... 36

iv

5.3.3 Brinell Hardness Test ......................................................................................... 36

5.3.4 Vickers Hardness Test ....................................................................................... 36

5.3.5 Knoop Hardness Test ......................................................................................... 37

5.3.6 Meyer Hardness Test ......................................................................................... 37

5.3.7 Destructive and Non-Destructive Test ............................................................... 38

5.4 Procedure ................................................................................................................... 38

5.5 Observations and Calculations .................................................................................. 38

5.6 Comments.................................................................................................................. 39

6 Lab Session – 6 ................................................................................................................ 40

6.1 Objective ................................................................................................................... 40

6.2 Apparatus .................................................................................................................. 40

6.3 Theory ....................................................................................................................... 40

6.3.1 Hardness Tests ................................................................................................... 40

6.3.2 Rockwell Hardness Test .................................................................................... 41

6.3.3 Brinell Hardness Test ......................................................................................... 41

6.3.4 Vickers Hardness Test ....................................................................................... 41

6.3.5 Knoop Hardness Test ......................................................................................... 42

6.3.6 Meyer Hardness Test ......................................................................................... 42

6.3.7 Destructive and Non-Destructive Test ............................................................... 42

6.4 Procedure ................................................................................................................... 43


6.6 Comments.................................................................................................................. 44

7 Lab Session – 7 ................................................................................................................ 45

7.1 Objective ................................................................................................................... 45

7.2 Apparatus .................................................................................................................. 45

7.3 Theory ....................................................................................................................... 45

7.3.1 Universal Testing Machine ................................................................................ 45

v

7.4 Procedure ................................................................................................................... 47


7.6 Graph ......................................................................................................................... 48

7.7 Comments.................................................................................................................. 48

8 Lab Session – 8 ................................................................................................................ 49

8.1 Objective ................................................................................................................... 49

8.2 Apparatus: ................................................................................................................. 49

8.3 Theory ....................................................................................................................... 49


8.4 Observations and Calculations: ................................................................................. 51

8.5 Graph ......................................................................................................................... 52

8.6 Comments.................................................................................................................. 52

9 Lab Session – 9 ................................................................................................................ 53

9.1 Objective ................................................................................................................... 53

9.2 Apparatus .................................................................................................................. 53

9.3 Theory ....................................................................................................................... 53


9.4 Procedure ................................................................................................................... 55


9.6 Graph ......................................................................................................................... 56

9.7 Comments.................................................................................................................. 56

10 Lab Session – 10 .............................................................................................................. 57

10.1 Objective ................................................................................................................ 57

10.2 Apparatus ............................................................................................................... 57

10.3 Theory .................................................................................................................... 57


10.4 Procedure ............................................................................................................... 59

vi

10.5 Observations and Calculations .............................................................................. 59

10.6 Graph ..................................................................................................................... 60

10.7 Comments .............................................................................................................. 60

11 Lab Session – 11 .............................................................................................................. 61

11.1 Objective ................................................................................................................ 61

11.2 Apparatus ............................................................................................................... 61

11.3 Theory .................................................................................................................... 61


11.4 Procedure ............................................................................................................... 63


11.6 Graph ..................................................................................................................... 64

11.7 Comments .............................................................................................................. 64

12 Lab Session – 12 .............................................................................................................. 65

12.1 Objective ................................................................................................................ 65

12.2 Apparatus ............................................................................................................... 65

12.3 Theory .................................................................................................................... 65


12.4 Procedure ............................................................................................................... 67


12.6 Graphs .................................................................................................................... 68

12.7 Comments .............................................................................................................. 68

13 Lab Session – 13 .............................................................................................................. 69

13.1 Objective ................................................................................................................ 69

13.2 Apparatus ............................................................................................................... 69

13.3 Theory .................................................................................................................... 69


13.4 Procedure ............................................................................................................... 71

vii


13.6 Graphs .................................................................................................................... 72

13.7 Comments .............................................................................................................. 72

14 Lab Session – 14 .............................................................................................................. 73

14.1 Objective ................................................................................................................ 73

14.2 Apparatus: .............................................................................................................. 73

14.3 Theory .................................................................................................................... 73


14.4 Procedure ............................................................................................................... 75

14.5 Observations and Calculations: ............................................................................. 75

14.6 Graphs .................................................................................................................... 76

14.7 Comments .............................................................................................................. 76

viii

LIST OF FIGURES

Figure 1-1: Propped Cantilever Beam ....................................................................................... 1

Figure 1-2: Cantilever beam ...................................................................................................... 2

Figure 1-3: Simply supported beam ........................................................................................... 2

Figure 1-4: Overhanging beams ................................................................................................. 3

Figure 1-5: (a) 2-sided overhanging beam (b) 1-sided overhanging beam ................................ 3

Figure 1-6: Propped Cantilever Beam ....................................................................................... 3

Figure 1-7: Fixed beam .............................................................................................................. 4

Figure 1-8: Continuous beam..................................................................................................... 4

Figure 1-9: Graph between deflection and loads ....................................................................... 5

Figure 1-10: Graph between deformation and load ................................................................... 6

Figure 2-1: Combine bending and torsion apparatus ................................................................. 8

Figure 2-2: Necking ................................................................................................................... 9

Figure 2-3: Bending ................................................................................................................. 10

Figure 2-4: Compression and stretchness of fibers .................................................................. 10

Figure 2-5: Bending moments in a beam ................................................................................. 10

Figure 2-6: Twisting ................................................................................................................ 11

Figure 2-7: Cross Sectional view of twisting ........................................................................... 11

Figure 2-8: Combined Bending and torsion............................................................................. 12

Figure 2-9: Bending vs. position of load in bending ............................................................... 13

Figure 2-10: Bending vs. position of load in torsion ............................................................... 15

Figure 2-11: Combined effect of bending & torsion vs. position of load ................................ 16

Figure 3-1: Quarter Circular Curved Bar Apparatus ............................................................... 18

Figure 3-2: Deflection of curved bars ...................................................................................... 19

Figure 3-3: Deflection of quarter circular bar .......................................................................... 22

Figure 3-4: Graph between load and vertical deflection .......................................................... 25

Figure 3-5: graph between load and horizontal deflection ...................................................... 25

Figure 4-1: Semi Circular Curved Bar Apparatus ................................................................... 27

Figure 4-2: Deflection of curved bar ....................................................................................... 28

Figure 4-3: Deflection of semi-circular bar ............................................................................. 30

Figure 4-4: Graph between horizontal deflection and load ..................................................... 32

Figure 4-5: Graph between vertical deflection and load .......................................................... 33

Figure 5-1: Rockwell Testing Machine ................................................................................... 35

ix

Figure 5-2: Specimen Material ............................................................................................... 35

Figure 5-3: Vickers Hardness Test .......................................................................................... 37

Figure 6-1: Rockwell Testing Machine ................................................................................... 40

Figure 6-2: Specimen Material Sample ................................................................................... 40

Figure 6-3: Vickers Hardness Test .......................................................................................... 42

Figure 7-1: Universal Testing Machine for mild steel ............................................................. 45

Figure 7-2: Necking Phenomenon for mild steel ..................................................................... 47

Figure 8-1: Universal Testing Machine for Aluminum ........................................................... 49

Figure 8-2: Necking Phenomenon for aluminum .................................................................... 51

Figure 9-1: Universal Testing Machine for copper .................................................................. 53

Figure 9-2: Necking Phenomenon for copper .......................................................................... 55

Figure 10-1: Universal Testing Machine for Brass.................................................................. 57

Figure 10-2: Necking Phenomenon for Brass .......................................................................... 59

Figure 11-1: Universal Testing Machine for plain steel alloy ................................................. 61

Figure 11-2: Necking Phenomenon for plain steel alloy ......................................................... 63

Figure 12-1: Universal Testing Machine for flat plate of polypropylene ................................ 65

Figure 12-2: Necking Phenomenon for flat plate of polypropylene ........................................ 67

Figure 13-1: Universal Testing Machine for concrete ............................................................. 69

Figure 13-2: Necking Phenomenon for concrete ..................................................................... 71

Figure 14-1: Universal Testing Machine for different beams ................................................. 73

Figure 14-2: Necking Phenomenon for different beams .......................................................... 75

x

LIST OF TABLES

Table 1-1: Variation in deflection with loads ............................................................................ 5

Table 1-2: Variation in deflection with loads ............................................................................ 6

Table 2-1: Position of load vs. reading of the dial gauge in bending ...................................... 13

Table 2-2: Position of load vs. reading of the dial gauge in torsion ........................................ 14

Table 2-3: Position of load vs. reading of dial gauge in both bending and torsion ................. 16

Table 3-1: Calculation of horizontal and vertical deflection with load ................................... 25

Table 4-1: Variation of deflection with load of a semi-circular beam ..................................... 32

Table 5-1: Diff. B/W Destructive& non-destructive test ......................................................... 38

Table 6-1: Diff. B/W Destructive& non-destructive test ......................................................... 43

Table 7-1: Calculations for Mild Steel..................................................................................... 48

Table 8-1: Calculations for Aluminium ................................................................................... 52

Table 9-1: Calculations for Copper.......................................................................................... 56

Table 10-1: Calculations for Brass .......................................................................................... 60

Table 11-1: Calculations for Plain Steel Alloy ........................................................................ 64

Table 12-1: Calculations for Flat Plate of Polypropylene ....................................................... 67

Table 13-1: Calculations for Concrete ..................................................................................... 71

Table 14-1: Calculations for different beams .......................................................................... 75

1

1 Lab Session – 1

1.1 Objective

To determine the deflection at the mid span of a propped cantilever beam given that 3-point

lads are acting on beam at equidistant from roller support using aluminum and brass and

compare their results.

1.2 Apparatus

Propped cantilever beam apparatus

Weights

Dial gauge

Vernier Caliper

Specimen

Hangers

Spanner

Figure 1-1: Propped Cantilever Beam

1.3 Summary of Theory

A beam is a structural element that is capable of withstanding load primarily by resisting

bending.

1.4 Classification of Beams

The beams may be classified in several ways, but the commonly used classification is based

on support conditions. On this basis the beams can be divided into six types:

i. Cantilever beams

ii. Simply supported beams

iii. Overhanging beams

iv. Propped beams

v. Fixed beams

vi. Continuous beams

2

1.4.1 Cantilever Beam

A beam having one end fixed and the other end free is known as cantilever beam, figure shows

a cantilever with end ‘A’ rigidly fixed into its supports, and the other end ‘B’ is free. The length

between A and B is known as the length of cantilever.

It has 3 reaction forces

Statically determinate

Figure 1-2: Cantilever beam

1.4.2 Simply Supported Beam

A beam having both the ends freely resting on supports is called a simply supported beam. The

reaction act at the ends of effective span of the beam. Figure show simply supported beams.

For such beams the reactions at the two ends are vertical. Such a beam is free to rotate at the

ends, when it bends.


Statically determinate

Figure 1-3: Simply supported beam

1.4.3 Overhanging Beams

A beam for which the supports re not situated at the ends and one or both ends extend over the

supports, is called an overhanging beam. Figure represents overhanging beams.


Statically indeterminate

3

Figure 1-4: Overhanging beams

(a) (b)

Figure 1-5: (a) 2-sided overhanging beam (b) 1-sided overhanging beam

1.4.4 Propped Cantilever Beams

A cantilever beam for which one end is fixed and other end is provided support, in order to

resist the deflection of the beam, is called a propped cantilever bema. A propped cantilever is

a statically indeterminate beam. Such beams are also called as restrained beams, as an end is

restrained from rotation.



Figure 1-6: Propped Cantilever Beam

1.4.5 Fixed Beams

A beam having its both the ends rigidly fixed against rotation or built into the supporting walls,

is called a fixed beam. Such a beam has four reaction components for vertical loading (i.e., a

vertical reaction and a fixing moment at both ends) figure shows the fixed beam.



4

Figure 1-7: Fixed beam

1.4.6 Continuous Beam

A beam having more than two supports, is called as continuous beam. The supports at the ends

are called as the end supports, while all the other supports are called as intermediate support.

It may or may not have overhang. It is statically indeterminate beam. In these beams there may

be several spans of same or different lengths figure shows a continuous beam.

It has more than 3 reaction forces


Figure 1-8: Continuous beam

1.5 Procedure

i. Measure the width and depth of the beam with the help of scale to find the moment of

inertia of the beam.

ii. Set the apparatus and put the required hangers at different points.

iii. Measure the distances of each hanger from the reference end.

iv. Set the deflection dial gauge at zero after putting the hangers.

v. Take the reading of deflection after putting the loads in the hangers

vi. Repeat the process for different loads

vii. Find the theoretical deflection and compare with the experimental values by showing

on a graph

1.6 Observations and Calculations (Aluminum)

Width of Beam = b = 25.4 mm

Depth of beam = d = 5.5 mm

Length of beam = 610mm

Moment of Inertia for rectangular metal bar = I = bd3/12= 504.46 𝑚𝑚4

5

Modulus of Elasticity = E = 70 GPa

Least count of dial gauge = 0.01mm

Table 1-1: Variation in deflection with loads

No.

of

Obs.

Loads (N) Deflection

(mm) δ exp

(Mean)

(mm)

δ th

=𝑊

192𝐸𝐼

(mm)

%age

Error W1 W2 W3 Loading

Un-

loading

1 1 1 1 0.24 0.4 0.32 0.29 9.3%

2 2 1 1 0.58 0.725 0.6525 0.589 9.6%

3 2 2 2 0.89 0.98 0.935 0.884 5.45%

4 4 2 2 1.16 1.27 1.215 1.17 3.70%

5 4 4 2 1.49 1.49 1.49 1.47 1.34%

1.7 Graph

On graph, plot the deflection against load for the theoretical & practical results. Draw the best-

fit straight lines through the points

Figure 1-9: Graph between deflection and loads

1.8 Observations and Calculations (Brass)

Width of Beam = b = 9 mm

Depth of beam = d = 18 mm

Moment of Inertia for rectangular metal bar = I = bd3/12= 4374 𝑚𝑚4

Modulus of Elasticity = E = 70 GPa

0.32

0.6525

0.935

1.215

1.49

0.29

0.589

0.884

1.17

1.47

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Loa

d (

N)

Loa

d (

N)

Deflection (mm)

Experimental Theoretical

6

Table 1-2: Variation in deflection with loads

Ob.

No

Loads (N) Deflection δ exp

(Mean) δ th =

𝟓𝟑𝟓𝟕 𝐖

𝐄𝐈

%age

Error W1 W2 W3 Loading Un-

loading

1 1 1 1 0.26 0.26 0.26 0.189245 27

2 2 1 1 0.47 0.47 0.47 0.37849 19

3 2 2 2 0.71 0.71 0.71 0.567734 20

4 4 2 2 0.98 0.98 0.98 0.756979 22

5 4 4 2 1.50 1.50 1.50 0.946224 36

1.8.1 Graph

On graph, plot the deflection against load for the theoretical & practical results. Draw the best-

fit straight lines through the points

Figure 1-10: Graph between deformation and load

1.9 Industrial Applications

The are some industrial applications of loading beams

Heavy duty beam trolleys

Concrete beam construction

Residential construction

Supporting the heavy loads

0

0.5

1

1.5

2

0 1 2 3 4 5 6

def

orm

atio

n

load

load vs def.

theoretical practical

7

1.10 Statistical Analysis

The mean value of the deflection is:

�̅� =0.26 + 0.47 + 0.71 + 0.98 + 1.50

5

= 0.784

Standard Deviation is:

𝑺. 𝑫 = √(0.784 − 0.26)2 + (0.784 − 0.47)2 + (0.784 − 0.71)2 + (0.784 − 0.98)2 + (0.784 − 1.50)²

5 − 1

= 0.482

1.11 Conclusion

We have learned a great deal about how the bending of a beam depends on the beam's load,

material properties, cross section, and manner of support. We use the static beam equation and

the ideas that we have explored as a basis for understanding the static deformations of more

complicated structures. Deflection of Aluminum is more as compared to brass.

1.12 Comments

There are a few valid reasons to make this experiment more accurate

Beam is made of aluminum, which shows more deflection

We can use steel beams which is harder material than aluminum

Steel beam will show the same properties for imposed loads which includes live

loads

It will also bear the wind loads, earthquake loads and snow loads

8

2 Lab Session – 2

2.1 Objective

To determine what levels of a combine bending and torsion cause elastic failure in

different materials and compare them with various theories of failure.

2.2 Apparatus

Bending & torsion apparatus

Weights

Hanger

Dial Gauge

Vernier Caliper

Spanner

Specimen

Figure 2-1: Combine bending and torsion apparatus

2.3 Theory

In this experiment we have to find out the effects of the bending and torsion on the

specimen under the observation. These things become the failure of materials. The

apparatus consists of a specimen “necked” between the base plate and the other end is

joined with the counter balanced circular loading plate.

9

Regular interval graduations on the loading plate allow a special hanger to locate. The

hanger enables us to measure the pure bending, the pure torsion or combined effect of

the bending and torsion, depending upon its position. The specimen deflection is

measured by a dial gauge mounted diametrically opposite load point.

This simple machine uses inexpensive test specimens made from round bar. The

specimen is clamped at one end to the base bracket and at the other to a counterbalanced

circular loading plate. This plate is graduated in 15° intervals. A special hanger enables

pure bending, pure torque or combined loads to be applied depending on the position of

the plate. The specimen deflection is measured by a dial gauge mounted diametrically

opposite the load point. In the event of a specimen failure safety is ensured by set screws.

2.3.1 Necking

Necking, in engineering or materials science, is a mode of tensile deformation where relatively

large amounts of strain localize disproportionately in a small region of the material. The

resulting prominent decrease in local cross-sectional area provides the basis for the name

"neck". Because the local strains in the neck are large, necking is often closely associated

with yielding, a form of plastic deformation associated with ductile materials, often metals or

polymers. The neck eventually becomes a fracture when enough strain is applied. Necking

results from an instability during tensile deformation when a material's cross-sectional area

decreases by a greater proportion than the material strain hardens.

Figure 2-2: Necking

https://en.wikipedia.org/wiki/Engineering

https://en.wikipedia.org/wiki/Materials_science

https://en.wikipedia.org/wiki/Deformation_(engineering)

https://en.wikipedia.org/wiki/Deformation_(mechanics)

https://en.wikipedia.org/wiki/Yield_(engineering)

https://en.wikipedia.org/wiki/Ductile

https://en.wikipedia.org/wiki/Fracture

https://en.wikipedia.org/wiki/Instability

https://en.wikipedia.org/wiki/Strain_hardening

10

2.3.2 Bending

Bending is defined as the reaction of the loading applied perpendicular to the longitudinal

axis of the element. In applied mechanics, bending (also known as flexure) characterizes the

behavior of a slender structural element subjected to an external load applied perpendicularly

to a longitudinal axis of the element.

Figure 2-3: Bending

Figure 2-4: Compression and stretchness of fibers

Figure 2-5: Bending moments in a beam

https://en.wikipedia.org/wiki/Applied_mechanics

https://en.wikipedia.org/wiki/Structural

https://en.wikipedia.org/wiki/Structural_load

11

2.3.3 Torsion

Twisting of the object due to the applied torque on the object. Its units are per square

pound. Torsion is the twisting of an object due to an applied torque. Torsion is expressed in

either the Pascal (Pa), an SI unit for newtons per square meter, or in pounds per square

inch (psi) while torque is expressed in newton meters (N·m) or foot-pound force (ft·lbf). In

sections perpendicular to the torque axis, the resultant shear stress in this section is

perpendicular to the radius.

Figure 2-6: Twisting

Figure 2-7: Cross Sectional view of twisting

2.3.4 Combine effect of bending and torsion

Applications of bending and torsion are very wide in our daily life routine, e.g. in shafts

of the engine, in the construction beams, in the loading machines these things are applied.

In designing many engineering apparatuses or in designing the weight lifting objects

https://en.wikipedia.org/wiki/Torque

https://en.wikipedia.org/wiki/Pascal_(unit)

https://en.wikipedia.org/wiki/SI

https://en.wikipedia.org/wiki/Pounds_per_square_inch

https://en.wikipedia.org/wiki/Pounds_per_square_inch

https://en.wikipedia.org/wiki/Newton_metre

https://en.wikipedia.org/wiki/Foot-pound_force

https://en.wikipedia.org/wiki/Shear_stress

12

these things are encounter and cause the failure of the system or the structure so that the

behavior of things or materials are necessary to deal. If it is ignoring, then many accidents

can happen which can damage or could be very dangerous for the life of the people.

Figure 2-8: Combined Bending and torsion

2.4 Bending

2.4.1 Procedure

Setup the apparatus on the horizontal table so can it would be able to hang the

weight to the base plate.

Set the dial gauge at the zero degree to the specimen for pure bending.

Place the hanger at the front and opposite to the dial gauge at zero degree.

Note the reading on the dial gauge.

Now start moving the hanger at a place next then to the 1st place by 15º and note

the reading keep doing this until minimum reading is obtained at the 90º.

Keep the thing carefully and take readings neatly.

2.4.2 Observations and Calculations

Weight applied = 5N

Self-weight of hanger= 10N

Total weight = 15N

Least count of dial gauge=0.01mm

Dial position = 0º

13

Table 2-1: Position of load vs. reading of the dial gauge in bending

Sr # Position of the load

(degree) Reading of the dial gauge

1 0º 0.39

2 15º 0.35

3 30º 0.27

4 45º 0.18

5 60º 0.13

6 75º 0.07

7 90º 0.01

Figure 2-9: Bending vs. position of load in bending

2.4.3 Statistical Analysis

The mean value of bending is given by:

M.D = �̅� =0.39+0.35+0.27+0.18+0.13+0.07+0.01

7

= 0.2 mm

𝑺. 𝑫 = √∑ (𝑿𝒊 − �̅�)𝒏

𝒊=𝟏

𝒏 − 𝟏

0.390.35

0.27

0.18

0.13

0.07

0.01

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 20 40 60 80 100

Ben

din

g (m

m)

Position of Load (degree)

Bending Vs Position of Load

14

Standard deviation is given as,

= √(0.39−0.2)2+(0.35−0.2)2+(0.27−0.2)2+(0.18−0.2)2+(0.13−0.2)2+(0.07−0.2)2+(0.01−0.2)2

6

= 0.058

2.5 Torsion

2.5.1 Procedure

Setup the things as in the 1st place with the difference that the dial gauge would be

at 90º rather than the zero degree.

Place the hanger likewise in the 1st case and note the reading of the dial gauge.

1st at the zero degree and then proceed to the 90º as in the 1st case.

In this case maximum reading would be at 90º and the minimum reading would

be at the 0 degree.


Weight applied = 5N


Total weight = 15N


Dial position = 90º

Table 2-2: Position of load vs. reading of the dial gauge in torsion

Sr # Position of load (degree) Reading of dial gauge

1 0º 0

2 15º 0.01

3 30º 0.05

4 45º 0.09

5 60º 0.11

6 75º 0.13

7 90º 0.14

15

Figure 2-10: Bending vs. position of load in torsion


The mean value of Torsion is given by:

M.D = �̅� =0+0.01+0.05+0.09+0.11+0.13+0.14

7

= 0.08 mm

𝑺. 𝑫 = √∑ (𝑿𝒊 − �̅�)𝒏

𝒊=𝟏

𝒏 − 𝟏

Standard deviation is given as:

= √(0−0.08)2+(0.01−0.08)2+(0.05−0.08)2+(0.09−0.08)2+(0.11−0.08)2+(0.13−0.08)2+(0.14−0.08)2

6

= 0.023

2.6 Combine Effect of bending and torsion

2.6.1 Procedure

Setup the apparatus as mention above and place the dial gauge at the 45º to the

specimen.

Place the hanger at the zero degree and note the reading

Now start moving the hanger from 0º to the 90º.

Take the readings at each angle.


Weight applied = 5N

0

0.01

0.05

0.09

0.11

0.130.14

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 20 40 60 80 100

Tors

ion

(mm

)


Torsion Vs Position of Load

16


Total weight = 15N


Dial position = 45º

Table 2-3: Position of load vs. reading of dial gauge in both bending and torsion

Sr # Position of the load

(degree)

Reading of dial gauge

(mm)

1 0º 0.28

2 15º 0.27

3 30º 0.24

4 45º 0.21

5 60º 0.19

6 75º 0.16

7 90º 0.12

Figure 2-11: Combined effect of bending & torsion vs. position of load


The mean value of Torsion is given by:

0.28 0.27

0.24

0.210.19

0.16

0.12

0

0.05

0.1

0.15

0.2

0.25

0.3

0 20 40 60 80 100

Co

mb

ined

eff

ect

of

Ben

din

g a

nd

To

rsio

n

(mm

)


Combined effect of bending and torsion Vs position of Load

17

M.D = �̅� =0.28+0.27+0.24+0.21+0.19+0.16+0.12

7

= 0.21mm

𝑺. 𝑫 = √∑ (𝑿𝒊 − �̅�)𝒏

𝒊=𝟏

𝒏 − 𝟏

Standard deviation is given as:

= √(0.28−0.21)2+(0.27−0.21)2+(0.24−0.21)2+(0.21−0.21)2+(0.19−0.21)2+(0.16−0.21)2+(0.12−0.21)2

6

= 0.024


Some applications in which you’re able to find combined bending and torsion include,

Drive Shafts design

Plate girders

2.8 Comments

Under pure torsion no bending moment is induced in the shaft but due to significant

Self Weight of shaft bending moment does induce in the shaft.

Under the pure bending or pure torsion, the maximum normal stress and the maximum

shear stress of the shafts are equal.

The normal stresses are zero under pure torsion hence the normal stresses in the

formulae of all failure theories are consider to be zero.

2.9 Conclusion

The value of the bending moment and torque are measured by applying different loads to the

apparatus. It is observed that our practical values are very close to the mean one with the

deviation of 0.21%. These deviations are caused by the vibrations and friction in the material.

18

3 Lab Session – 3

3.1 Objective

To analyze the variation in experimental and theoretical deflection both (horizontal and

vertical) of a quarter circular beam.

3.2 Apparatus

i. Curved Bar Apparatus

ii. Weight

iii. Quarter circular beam apparatus

iv. Dial gauge

v. Vernier Caliper

Figure 3-1: Quarter Circular Curved Bar Apparatus

3.3 Summary of Theory:

3.3.1 Curved bars / deflection of curved bars

A body whose geometric shape is formed by the motion in space of a plane figure is called the

cross section of the curved beam); its center of gravity always follows a certain curve (the axis),

and the plane of the figure is normal to the curve. A distinction is made between curved beams

with constant cross section (for example, the link of a chain composed of oval or circular rings)

and with variable cross section (for example, the hook of a crane) and between plane beams

(with a plane axis) and three-dimensional beams (with a three-dimensional axis). A special

variety of curved beam is the naturally twisted curved beam, whose plane cross-sectional figure

moves along its axis and simultaneously rotates around a tangent to the axis (for example, the

blade of an aircraft propeller or fan).

19

The design of a plane curved beam (Figure 1) with a symmetrical cross section (the axis of

symmetry lies in the plane of curvature) taking into account the effect of a load lying in the

plane of symmetry consists in the determination of stresses normal to the cross section

according to the formula.

𝜎 =𝑁

𝐹+

𝑀𝑦

𝑆𝑧𝜌

where F is the area of the cross section, N is the longitudinal force, M is the bending moment

in the cross section defined with respect to the axis Z0 passing through the center of gravity of

the cross section (C), y is the distance from the fiber being examined to the neutral axis z, p is

the radius of curvature of the fiber being examined, and Sz = Fy0 is the static moment of the

cross-sectional area with respect to the axis z. The displacement Y0 of the neutral axis relative

to the center of curvature of the curved beam is always directed toward the center of curvature

of the curved beam and is usually determined from special tables. For a circular cross section,

Y0 ≈ d2/16R; for a rectangular cross section, Y ≈ h2/12R (R is the radius of curvature of the

axis of the curved beam; d and h are the diameter and height of the cross section of the beam,

respectively). Normal stresses in a curved beam have their maximum values (in absolute

magnitudes) near the concave edge of a beam and vary in the cross section according to a

hyperbolic law. For small curvatures (R > 5h) the determination of normal stresses can be made

in the same way as for a straight beam.

Figure 3-2: Deflection of curved bars

3.3.2 Castiglione’s Theorem

Determining the deflection of beams typically requires repeated integration of singularity

functions. Castigliano’s Theorem lets us use strain energies at the locations of forces to

20

determine the deflections. The Theorem also allows for the determining of deflections for

objects with changing cross-sectional areas.

Whenever a load is applied to a spring it will show some deflection. This deflection is directly

related to the force applied on the spring to produce that deflection. The force deflection

relationship is most conveniently obtained using Castigliano's theorem. Which is stated as

“When forces act on elastic systems subject to small displacements, the displacement

corresponding to any force collinear with the force is equal to the partial derivative to the

total strain energy with respect to that force.”

And It is given as

𝛿 =𝜕𝑈

𝜕𝑃

In order to derive a necessary formula which governs the behavior of springs, consider a closed

coiled spring subjected to an axial load W.

Let,

W = Axial Load

D = Mean Coil Diameter

d = Diameter of Spring Wire N = Number of Active Coils

l = Length of Spring

Wire = πDN

G = Modulus of Rigidity

∆ = Deflection of Spring

Φ = Angle of twist

In 1879, Alberto Castigliano’ an Italian railroad engineer, published a book in which he

outlined a method for determining the displacement / deflection & slope at a point in a body.

This method which referred to Catigliano’s Theorem is applied to the bodies, having constant

temperature & material (homogeneous) with linear elastic behavior.

It states that “The derivative of the strain energy with respect to the applied load gives the

deformation corresponding to that load. For a helical spring, the partial derivative of the strain

energy w.r.t. the applied load gives the deflection in the spring i.e. ∂U / ∂W = deflection.

Consider a helical compression spring made up of a circular wire and subjected to axial load

W as shown in the figure above.

Strain Energy is given by:

U = ½ T * Φ (ii)

Whereas,

21

T = ½ W * D (iii)

Φ = Tl / JG (iv)

(From Torsion formula) putting the values from eqs. # (i), (iii) & (iv) in eq. # (ii) and

simplifying, we get;

T= 4 W2D 3N / d4G (v)

Now applying the castigliano’ theorem by taking the partial derivative of the strain energy with

respect to the applied load

∂U / ∂W = ∆ = 8 WD3N / d4G (v)

W / ∆ = d4G / 8 D3N

3.3.3 Derivation of the Castigliano’s Theorem for Quarter Circular Bar

The general expression of Castigliano’s theorem is as follows:

𝛿 = ∫ 𝑀/𝐸𝐼

𝑆

0

∗𝑑𝑀

𝑑𝑊∗ 𝑑𝑠 =>

1

𝐸𝐼∗ ∫ 𝑀

𝑆

0

∗𝑑𝑀

𝑑𝑊∗ 𝑑𝑠

where

M = is the moment induced by the force of loading,

E = is the elastic modulus of the beam material,

I = is the moment of inertia of the beam,

dM/dW = is the change in moment with respect to the force of loading and

ds = is the finite quantity of the beam over which integration is to take place. Because the

modulus = E and the moment of inertia = I am constants, they are factored out of the integral.

The work of deformation, or the moment, can be expressed as the product of the loading force,

P, the radius from the center of curvature of the beam R

and the sine of the angle of curvature.

The moment can be expressed by the following equation:

𝑀 = 𝑃𝑅𝑠𝑖𝑛𝜃

The integrating factor ds of the general Castigliano equation can be expressed as follows:

𝑑𝑠 = 𝑅𝑑𝜃

The partial derivative of the work of deformation with respect to the component of the force is

expressed as a function of the radius of the beam and angle of the deflected beam. For the

vertical deflection, the partial derivative is written as:

(𝑑𝑀

𝑑𝑊)

𝑉= 𝑅𝑠𝑖𝑛𝜃

and for the horizontal deflection of a curved beam, the partial derivative is written as:

22

(𝑑𝑀

𝑑𝑊)

𝐻= 𝑅(1 − 𝑐𝑜𝑠𝜃)

Figure 3-3: Deflection of quarter circular bar

The calculations for the vertical and horizontal deflection of the davit differs lightly from those

of the semicircular beam. The davit consists of a quarter circle curved beam and a straight leg

that connects to the base as seen in Figure 2. This means that the calculations of deflection

must be broken into two parts: one integral for the curved section of the beam and another for

the straight leg of the beam. The integration of the curved section of the davit is bound by zero

and π/2 because it is a quarter -circle and the integration of the leg is bound by zero at the base

of the beam and the length L of the straight segment of the beam.

To calculate the vertical deflection caused by a force of loading for a davit, the general equation

of Castigliano’s theorem is modified to account for the straight segment of the beam. Substitute

Equations 2, 3 and 4 into the general Castigliano equation and append an integral that expresses

the moment endured by the straight segment.

∆𝑉 =1

𝐸𝐼∗ [∫ 𝑃𝑅𝑠𝑖𝑛𝜃 ∗ 𝑅𝑠𝑖𝑛𝜃 ∗ 𝑅𝑑𝜃 + ∫ 𝑃𝑅 ∗ 𝑅𝑑𝑦

𝐿

0

𝜋2

0

]

Factoring out the constants P and R yields the following expression:

∆𝑉 =1

𝐸𝐼∗ [𝑃𝑅3 ∫ sin2 𝜃𝑑𝜃 + 𝑃𝑅2 ∫ 𝑑𝑦]

𝐿

0

𝜋/2

0

Integrating with respect to theta and the y direction yields the following expression

23

∆𝑉 =1

𝐸𝐼∗ [𝑃𝑅3 (

𝜋

4) + 𝑃𝑅2𝐿]

and can be tidied up a little and the equation for the vertical deflection of the davit can be

written as follows:

∆𝑉 =[𝜋𝑃𝑅3]

[4𝐸𝐼]+

[𝑃𝑅2𝐿]

[𝐸𝐼]

But as in this case we are neglecting the length l of the straight segment of the beam because

in our apparatus the beam is tied from the starting section of the quarter circular beam so L =

0 which will yield us the following final relation

∆𝑉 =[𝜋𝑃𝑅3]

[4𝐸𝐼]

The straight segment of the davit must be accounted for in much the same way as it was for the

vertical deflection in the formulation of the horizontal deflection calculation.

∆𝐻 =1

𝐸𝐼∗ [∫ [𝑃𝑅𝑠𝑖𝑛𝜃 + 𝐻𝑅(1 − 𝑐𝑜𝑠𝜃)][𝑅(1 − 𝑐𝑜𝑠𝜃)]𝑅𝑑𝜃

𝜋/2

0

+ ∫ [𝑃𝑅 + 𝐻(𝑅 + 𝑦][𝑅 + 𝑦]𝑑𝑦𝐿

0

Substituting Equations 2, 3 and 5 into the general expression of Castigliano’s theorem and

appending an integral to describe the deflection in the straight segment of the davit yields the

following: Factoring out the constants P and R and letting the dummy variable H equal zero,

previous Equation becomes the following:

∆𝐻 =1

𝐸𝐼∗ [𝑃𝑅3 ∫ 𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃𝑑𝜃 + 𝑃 ∫ (𝑅2 + 𝑅𝑦)𝑑𝑦

𝐿

0

𝜋/2

0

Integrating for the curvature and straight segment yields the following expression

∆𝐻 =1

𝐸𝐼∗ [

𝑃𝑅3

2+ 𝑃𝑅2𝐿 +

𝑃𝑅𝐿2

2]

Distributing the modulus of elasticity E and moment of inertia I into this Equation yields

∆𝐻 = [𝑃𝑅3

2𝐸𝐼] + [

𝑃𝑅2𝐿

𝐸𝐼] + [

𝑃𝑅𝐿2

2𝐸𝐼]

Tidied up a little further, the equation for the horizontal deflection of a davit can be written as

follows

24

∆𝐻 = [𝑃𝑅3

2𝐸𝐼] + [

𝑃𝑅𝐿

2𝐸𝐼] [2𝑅 + 𝐿]

But as in this case we are neglecting the length l of the straight segment of the beam because

in our apparatus the beam is tied from the starting section of the quarter circular beam so L =

0 which will yield us the following final relation

∆𝐻 = [𝑃𝑅3

2𝐸𝐼]

3.4 Procedure:

i. Adjust the quarter circular bar.

ii. Attach two dial gauges for finding vertical as well as horizontal deflection

iii. Load the bar for number of times by an equal amount of 1N each time and note the

corresponding readings from dial gauges attached to the apparatus, for vertical and

horizontal deflection.

iv. Multiply those observations with the least count of the dial gauges and note out the

final deflections

3.5 Observations & Calculations:

Radius of curved bar = R = 100mm

Thickness of the bar = d = 3.175 mm

Moment of Inertia= I= 3.3 x 10-11 m4

Modulus of Elasticity = E= 207 GN/m2

3.5.1 Specimen calculations

P = 1 N

R = 100mm = 0.1m

E = 207 GN/m2

I = 3.3 x 10-11 m4

Δ H = PR3 / 2EI = 1N * (0.1m)3 / ( 2 * 207 * 109 N/m2 * 3.3 x 10-11 m4 )

Δ H = 0.0000731 m = 0.0731 mm

Δ V = πPR3 / 4EI = 3.14 * 1N * (0.1m)3 / ( 4 * 207 * 109 N/m2 * 3.3 x 10-11 m4 )

Δ V = 0.0001149 m = 0.1149 mm

3.5.2 Graph

On graph I plot the deflection against load for horizontal & vertical deflection for the theoretical

& practical results. I Draw the best fit straight lines through the points.

25

Figure 3-4: Graph between load and vertical deflection

Figure 3-5: graph between load and horizontal deflection

Table 3-1: Calculation of horizontal and vertical deflection with load

Sr. No.

Load

W

(N)

Dial gauge

reading

Experimental

deflection

(mm)

Theoretical deflection

(mm)

H V Δh Δv Δh= wr3/2ei Δv=πwr3/4ei

1 1 4 6 0.04 0.06 0.073 0.114

2 2 13 16 0.13 0.16 0.146 0.229

3 3 17 22 0.17 0.22 0.219 0.344

4 4 24 30 0.24 0.30 0.292 0.459

5 5 30 38 0.30 0.38 0.365 0.574

26


𝑋𝑎𝑣 =1

𝑛(𝑥1 + 𝑥2 + 𝑥3)

𝑆𝑥 = √1

𝑛 − 1((𝑥1 − 𝑥𝑎𝑣)2 + (𝑥2 − 𝑥𝑎𝑣)2 + (𝑥3 − 𝑥𝑎𝑣)2)

δHavg = 0.12166

δVavg = 1.83

SH = 0.102

SV =1.65

3.7 Industrial Applications:

i. Chains

ii. Hooks

iii. Loops

iv. Bridges

3.8 Comments

i. Vertical Deflection are very high as compared to the horizontal deflections.

ii. The Reason of large vertical deflections is the weight is being applied vertically.

iii. The gravity is also acting in this direction.

iv. Applying a horizontal load will cause deflections in horizontal deflections more

prominent

27

4 Lab Session – 4

4.1 Objective

To find out the horizontal and vertical deflection of semicircular beam loaded by vertical load

using the curved beam apparatus.

4.2 Apparatus

Curved Bar Apparatus

Weight

Semicircular beam apparatus

Dial gauge

Vernier Caliper.

Figure 4-1: Semi Circular Curved Bar Apparatus

4.3 Summary of Theory:

4.3.1 Curved Bars

A body whose geometric shape is formed by the motion in space of a plane figure (called the

cross section of the curved beam); its center of gravity always follows a certain curve (the axis),

and the plane of the figure is normal to the curve. A distinction is made between curved beams

with constant cross section (for example, the link of a chain composed of oval or circular rings)

and with variable cross section (for example, the hook of a crane) and between plane beams

(with a plane axis) and three-dimensional beams (with a three-dimensional axis). A special

variety of curved beam is the naturally twisted curved beam, whose plane cross-sectional figure

moves along its axis and simultaneously rotates around a tangent to the axis (for example, the

blade of an aircraft propeller or fan).

28

The design of a plane curved beam (Figure 1) with a symmetrical cross section (the axis of

symmetry lies in the plane of curvature) taking into account the effect of a load lying in the

plane of symmetry consists in the determination of stresses normal to the cross section

according to the formula.

where F is the area of the cross section, N is the longitudinal force, M is the bending moment

in the cross section defined with respect to the axis Z0 passing through the center of gravity of

the cross section (C), y is the distance from the fiber being examined to the neutral axis z, p is

the radius of curvature of the fiber being examined, and Sz = Fy0 is the static moment of the

cross-sectional area with respect to the axis z. The displacement Y0 of the neutral axis relative

to the center of curvature of the curved beam is always directed toward the center of curvature

of the curved beam and is usually determined from special tables. For a circular cross section,

Y0 ≈ d2/16R; for a rectangular cross section, Y ≈ h2/12R (R is the radius of curvature of the

axis of the curved beam; d and h are the diameter and height of the cross section of the beam,

respectively). Normal stresses in a curved beam have their maximum values (in absolute

magnitudes) near the concave edge of a beam and vary in the cross section according to a

hyperbolic law. For small curvatures (R > 5h) the determination of normal stresses can be made

in the same way as for a straight beam.

Figure 4-2: Deflection of curved bar

4.4 Derivation of formulae

4.4.1 Derivation of the Castigliano’s Theorem for Semi-Circular Bar

The general expression of Castigliano’s theorem is as follows:

29

Where;

M ► is the moment induced by the force of loading,

E ► is the elastic modulus of the beam material,

I ► is the moment of inertia of the beam,

dM/dW ► is the change in moment with respect to the force of loading and

ds ► is the finite quantity of the beam over which integration is to take place. Because the

modulus ► E and the moment of inertia ►I are constants, they are factored out of the integral.

The work of deformation, or the moment, can be expressed as the product of the loading force,

P, the radius from the center of curvature of the beam R

and the sine of the angle of curvature.

The moment can be expressed by the following equation:

The integrating factor ds of the general Castigliano equation can be expressed as follows:

The partial derivative of the work of deformation with respect to the component of the force is

expressed as a function of the radius of the beam and angle of the deflected beam. For the

vertical deflection, the partial derivative is written as:

and for the horizontal deflection of a curved beam, the partial derivative is written as:

To calculate the vertical deflection of a semicircular

beam, substitute Equations 2,3 and 4 into the general expression of Castigliano’s theorem

(Equation 1). The integration is bounded by zero and π because the beam is a semicircle. This

process will yield the following equation.

30

Figure 4-3: Deflection of semi-circular bar

The loading force P, the radius R, the elastic modulus E and the moment of inertia I, are all

constants and can be factored out of the integral. Integrating with respect to theta yields the

following equation for the vertical deflection of semicircular beam:

And thus, the final relation will be

To calculate the horizontal deflection of a semicircular beam, a dummy variable H must m

employed as seen in Figure 1. H represents a fictitious loading force in the horizontal direction.

Inserting the dummy variable allows for the integration in the horizontal direction. Substituting

Equations 2, 3 and 5 into the general expression of Castigliano’s theorem yields the following

expression.

Factoring out the constants P and R, and letting H equal zero, this equation becomes:

Integrating this equation with respect to theta yields the equation for the horizontal deflection

of a curved beam.

And thus, the final relation will be

31

4.5 Procedure

i. Adjust the semicircular bar.

ii. Attach two dial gauges for finding vertical as well as horizontal deflection

iii. Load the bar for number of times by an equal amount of 1N each time and note the

corresponding readings from dial gauges attached to the apparatus, for vertical and

horizontal deflection.

iv. Multiply those observations with the least count of the dial gauges and note out the

final deflections

4.6 Observations & Calculations

Radius of curved bar = R = 100mm

Width of the bar = b = 3.3 x 10-11 m4

Thickness of the bar = d = 207 GN/m2

Modulus of Elasticity = E= 12.7mm

Moment of Inertia= I= 3.175 mm

4.6.1 Specimen calculations

P = 1 N

R = 100mm = 0.1m

E = 207 GN/m2

I = 3.3 x 10-11 m4

Δ H = PR3 / 2EI = 1N * (0.1m)3 / ( 2 * 207 * 109 N/m2 * 3.3 x 10-11 m4 )

Δ H = 0.0000731 m = 0.0731 mm

Δ V = πPR3 / 4EI = 3.14 * 1N * (0.1m)3 / ( 4 * 207 * 109 N/m2 * 3.3 x 10-11 m4 )

Δ H = 0.0001149 m = 0.1149 mm

32

Table 4-1: Variation of deflection with load of a semi-circular beam

Sr.

No.

LOAD

W

(N)

Dial Gauge

Reading

Experimental

Deflection

(mm)

Theoretical Deflection

(mm)

H V δH δV δH=

7WR3/4EI

δV==

ΠWR3/2EI

1 1 1 2 0.01 0.02 0.02 0.02

2 2 9 19 0.09 0.19 0.05 0.04

3 3 13 26 0.13 0.26 0.07 0.06

4 4 19 35 0.19 0.35 0.1 0.08

5 5 20 49 0.20 0.49 0.13 0.1

4.6.2 Graph

On graph plot the deflection against load for horizontal & vertical deflection for the theoretical

& practical results. Draw the best fit straight lines through the points.

Figure 4-4: Graph between horizontal deflection and load

0

0.05

0.1

0.15

0.2

0.25

0 1 2 3 4 5 6

δH

in m

m

Load in N

Graph between Load and δH ( Exp. + Theo)

Experimental

Theorectical

33

Figure 4-5: Graph between vertical deflection and load


X Δh =0.27

X Δv =0.18

S Δh= 0.26

S Δv=0.15


Chains.

Links.

4.9 Comments

From analytical values it was observed that, as the load increases horizontal and

vertical deflections both increase significantly.

The horizontal deflection is more compared to vertical deflection.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6

δV

in m

m

Load in N

Graph between Load and δV ( Exp. + Theo)

Experimental

Theorectical

34

With using the Castiglia no’s Theorem Method in calculate the bar deflection, it easier

if compared with other method.

The result of deflection value is not far between experimental and theoretical.

This experiment is successful that is the deflection bar formula according to the

Castiglia no’s Theorem can be approved it punctuality for getting the beam deflection

35

5 Lab Session – 5

5.1 Objective

To study the Rockwell hardness testing machine and perform Rockwell hardness test on HRA

and HRC materials by using diamond indenter.

5.2 Apparatus

Rockwell Hardness Testing machine

Specimen (Material Hardness Code 91.3 HR15N N 122-703)

Figure 5-1: Rockwell Testing Machine

Figure 5-2: Specimen Material

5.3 Theory

5.3.1 Hardness Tests

There are mainly 5 hardness tests that are used for non-destructive testing of materials.

Hardness test is performed to evaluate the properties of materials such as strength,

36

ductility and wear resistance and so it helps to determine whether a material treatment

or use of that material is suitable for the specific purpose.

5.3.2 Rockwell Hardness Test

Rockwell hardness test is the simplest and the most cost-effective test which involves

applying a specific load on material using an indenter and measuring how far indenter

penetrates.

Working

Rockwell hardness test uses a conical diamond as an indenter. Initially minor load is

applied. An additional major force is then applied for a predetermined period known as

Dwell Period and then reduced to minor load state.

5.3.3 Brinell Hardness Test

The Brinell hardness test method as used to determine Brinell hardness, is defined in

ASTM E10. Most commonly it is used to test materials that have a structure that is too

coarse or that have a surface that is too rough to be tested using another test method,

e.g., castings and forgings.

Working

Brinell testing often use a very high-test load (3000 kgf) and a 10mm diameter indenter

so that the resulting indentation averages out most surface and sub-surface

inconsistencies.

The Brinell method applies a predetermined test load (F) to a carbide ball of fixed

diameter (D) which is held for a predetermined time period and then removed. The

resulting impression is measured with a specially designed Brinell microscope or

optical system across at least two diameters – usually at right angles to each other and

these results are averaged (d). Although the calculation can be used to generate the

Brinell number, most often a chart is then used to convert the averaged diameter

measurement to a Brinell hardness number.

5.3.4 Vickers Hardness Test

The Vickers hardness test method consists of indenting the test material with a diamond

indenter, in the form of a right pyramid with a square base and an angle of 136 degrees

between opposite faces subjected to a load of 1 to 100 kgf.

Working

The full load is normally applied for 10 to 15 seconds. The two diagonals of the

indentation left in the surface of the material after removal of the load are measured

https://www.hardnesstesters.com/products/brinell-optical-systems


37

using a microscope and their average calculated. The area of the sloping surface of the

indentation is calculated. The Vickers hardness is the quotient obtained by dividing the

kgf load by the square mm area of indentation.

Figure 5-3: Vickers Hardness Test

5.3.5 Knoop Hardness Test

The Knoop hardness test method, also referred to as a microhardness test method, is

mostly used for small parts, thin sections, or case depth work.

Working

The Vickers method is based on an optical measurement system. The Microhardness

test procedure, ASTM E-384, specifies a range of light loads using a diamond indenter

to make an indentation which is measured and converted to a hardness value. It is very

useful for testing on a wide type of materials as long as test samples are carefully

prepared.

5.3.6 Meyer Hardness Test

The Meyer hardness test is a rarely used hardness test based upon projected area of an

impression. This is a more fundamental measurement of hardness than other hardness

tests which are based on the surface area of an indentation. The principle behind the

test is that the mean pressure required to test the material is the measurement of the

hardness of the material. The mean pressure is calculated by dividing the load by the

projected area of the indentation. The result is called the Meyer hardness, which has

units of megapascals (MPa).

https://www.hardnesstesters.com/products/hardness-testing-accessories/indenters

https://en.wikipedia.org/wiki/Hardness

https://en.wikipedia.org/wiki/Projected_area

https://en.wikipedia.org/wiki/Pascal_(pressure)

38

5.3.7 Destructive and Non-Destructive Test

Table 5-1: Diff. B/W Destructive& non-destructive test

Non-Destructive Test Destructive Test

Used for finding out defects of material Used for finding out the properties of

the material

Load is not applied on the material Load is applied on the material

No load applications, so no chance for

material damage

Due to load application, material get

damaged

No requirement of special equipment’s Special equipment’s are required

Non expensive Expensive

Less skill Skill is required

e.g. dye penetrate test, ultrasonic,

radiography, etc.

e.g. tensile test, compression test,

hardness test, etc.

5.4 Procedure

I chose one of the given specimens and noted down the code written on it.

I put the sample material on the anvil and let the instructor do the rest.

Instructor put the sample material on the anvil at right place and raised the anvil till the

indenter just touched the material.

Instructor set the Rockwell Measuring Scale as per according to material which was

HR15N and then set the dwell period up-to 6 seconds.

Then exchange scale was set which was HRC.

Then the instructor pressed the start button and testing started.

One of the group students recorded the video of whole test to measure the value of

Major Load, Minor Load, exchange scale value.

Then, the whole group noted down the value of major load and minor load.

We also noted down the value of Hardness of material.

5.5 Observations and Calculations

Sr# Rockwell

Measuring scale Indenter

Major

Load(kgf)

Minor

Load(kgf) Hardness

Exchange

Scale

Value

39

1. HR15N DIAMOND

TIP 15 2.98 81 41.2 HRC

2. HRB DIAMOND

TIP 100 10 91.8

3. HR30T DIAMOND

TIP 30 2.98 1.8

4. HR30N DIAMOND

TIP 30 3 53.1 32.8 HRC

6. HRC DIAMOND

TIP 150 10 2.0 0 HRB

5.6 Comments

Different materials have different hardness.

Value of hardness is measured in different scales.

One scale value can be obtained in exchange scale.

40

6 Lab Session – 6

6.1 Objective

To study the Rockwell hardness testing machine and perform Rockwell hardness test on HRB

material by using Ball indenter.

6.2 Apparatus

Rockwell Hardness Testing machine

Specimen

Figure 6-1: Rockwell Testing Machine

Figure 6-2: Specimen Material Sample

6.3 Theory

6.3.1 Hardness Tests

There are mainly 5 hardness tests that are used for non-destructive testing of materials.

Hardness test is performed to evaluate the properties of materials such as strength,

41

ductility and wear resistance and so it helps to determine whether a material treatment

or use of that material is suitable for the specific purpose.

6.3.2 Rockwell Hardness Test

Rockwell hardness test is the simplest and the most cost-effective test which involves

applying a specific load on material using an indenter and measuring how far indenter

penetrates.

Working

Rockwell hardness test uses a conical diamond as an indenter. Initially minor load is

applied. An additional major force is then applied for a predetermined period known as

Dwell Period and then reduced to minor load state.

6.3.3 Brinell Hardness Test

The Brinell hardness test method as used to determine Brinell hardness, is defined in

ASTM E10. Most commonly it is used to test materials that have a structure that is too

coarse or that have a surface that is too rough to be tested using another test method,

e.g., castings and forgings.

Working

Brinell testing often use a very high-test load (3000 kgf) and a 10mm diameter indenter

so that the resulting indentation averages out most surface and sub-surface

inconsistencies.

The Brinell method applies a predetermined test load (F) to a carbide ball of fixed

diameter (D) which is held for a predetermined time period and then removed. The

resulting impression is measured with a specially designed Brinell microscope or

optical system across at least two diameters – usually at right angles to each other and

these results are averaged (d). Although the calculation can be used to generate the

Brinell number, most often a chart is then used to convert the averaged diameter

measurement to a Brinell hardness number.

6.3.4 Vickers Hardness Test

The Vickers hardness test method consists of indenting the test material with a diamond

indenter, in the form of a right pyramid with a square base and an angle of 136 degrees

between opposite faces subjected to a load of 1 to 100 kgf.

Working

The full load is normally applied for 10 to 15 seconds. The two diagonals of the

indentation left in the surface of the material after removal of the load are measured



42

using a microscope and their average calculated. The area of the sloping surface of the

indentation is calculated. The Vickers hardness is the quotient obtained by dividing the

kgf load by the square mm area of indentation.

Figure 6-3: Vickers Hardness Test

6.3.5 Knoop Hardness Test

The Knoop hardness test method, also referred to as a microhardness test method, is

mostly used for small parts, thin sections, or case depth work.

Working

The Vickers method is based on an optical measurement system. The Microhardness

test procedure, ASTM E-384, specifies a range of light loads using a diamond indenter

to make an indentation which is measured and converted to a hardness value. It is very

useful for testing on a wide type of materials as long as test samples are carefully

prepared.

6.3.6 Meyer Hardness Test

The Meyer hardness test is a rarely used hardness test based upon projected area of an

impression. This is a more fundamental measurement of hardness than other hardness

tests which are based on the surface area of an indentation. The principle behind the

test is that the mean pressure required to test the material is the measurement of the

hardness of the material. The mean pressure is calculated by dividing the load by the

projected area of the indentation. The result is called the Meyer hardness, which has

units of megapascals (MPa).

6.3.7 Destructive and Non-Destructive Test

https://www.hardnesstesters.com/products/hardness-testing-accessories/indenters

https://en.wikipedia.org/wiki/Hardness

https://en.wikipedia.org/wiki/Projected_area

https://en.wikipedia.org/wiki/Pascal_(pressure)

43

Table 6-1: Diff. B/W Destructive& non-destructive test

Non-Destructive Test Destructive Test

Used for finding out defects of material Used for finding out the properties of

the material

Load is not applied on the material Load is applied on the material

No load applications, so no chance for

material damage

Due to load application, material get

damaged

No requirement of special equipment’s Special equipment’s are required

Non expensive Expensive

Less skill Skill is required

e.g. dye penetrate test, ultrasonic,

radiography, etc.

e.g. tensile test, compression test,

hardness test, etc.

6.4 Procedure

I chose one of the given specimens and noted down the code written on it.

I put the sample material on the anvil.

I placed the sample material on the anvil at right place and raised the anvil till the

indenter just touched the material.

I set the Rockwell Measuring Scale as per according to material which was HRB and

then set the dwell period up-to10 seconds.

Then exchange scale was set which was HRB.

I repeated the same procedure for 8 second dwell period and HRC exchange scale.

Then, the whole group noted down the value of major load and minor load.

We also noted down the value of Hardness of material.

44


Sr# Rockwell

Measuring scale Indenter

Major

Load(kgf)

Minor

Load(kgf) Hardness

Exchange

Scale

Value

1. HRB Ball

Indenter 100 10 77.4 77.4 HRB

2. HRB Ball

Indenter 100 10 78.0 0.00 HRC

6.6 Comments

Different materials have different hardness.

Value of hardness is measured in different scales.

One scale value can be obtained in exchange scale.

Ball indenter is basically used for soft materials.

Diamond tip indenter is basically used for harder materials.

Variation in the value of hardness is due to inexperience.

45

7 Lab Session – 7

7.1 Objective

To perform tensile test on a specimen of mild steel (MS) by using universal testing machine

(UTM)

7.2 Apparatus

Specimen (Plain Steel)

Universal Testing Machine Control Unit

Computer System to compile the results

7.3 Theory

7.3.1 Universal Testing Machine

A machine used to test specimens for tensile strength, compressive strength, shear strength

and to perform bend test along other important laboratory tests. The primary use of the testing

machine is to create the stress strain diagram. Universal testing machine (UTM) is called so

because of the versatility of its application. The following tests can be performed with it

Tension Test

Compression Test

Bending Test

Figure 7-1: Universal Testing Machine for mild steel

46

7.3.1.1 Parts of Universal Testing Machine

The Universal Testing Machine consists of two main parts

1. The loading unit

2. The control panels

Loading Unit

In this unit actual loading of the specimen takes place - consists of three cross heads namely

upper head, middle head and lower head. Using appropriate cross heads tensile, compressive,

shear, bending load with the help of different attachment can be applied. Loading unit of a

UTM consists of:

1. Upper cross head

2. Lower cross head

3. Table Upper Cross Head

It is used to clamp testing specimen from top

Lower Cross Head

It is used to clamp testing specimen from below

Table

It is used to place the specimen, used for compression test

Elongation Scale

An elongation scale, which measures the relative movement between the lower table and the

lower cross-head, is also provided with the loading unit.

Control Unit

These include the electric control devices, the hydraulic control devices and the load

indicating devices.

Electric Control Devices

These are in the form of four switches set on the left side of the panel face. The upper and

lower push switches are for moving the lower cross-head up and down respectively. The

remaining two are the ON and OFF switches for the hydraulic pump.

Hydraulic Control Devices

These are a pair of control valves set on the table or the control panel.

These are:

The right control valves

The left control valves

47

The right control valve is the inlet valve and the left control valve is the return valve.

Load indicating Devices

A means of providing the test result is needed. Some older machines have dial or digital

displays and chart recorder. Many newer machines have a computer interface for analysis

and printing.

Extensometer

An instrument used to measure elongation in the material

Necking

At a certain maximum value of the load the diameter of a portion of the specimen begins to

decrease, because of local instability. This phenomenon is known as necking. To visualize the

necking effect of a material, a tensile test is conducted on a specimen of the material. The test

specimen is placed in a testing machine which is used to apply a centric load P. At a certain

maximum stress, the diameter of a portion of the specimen start to decrease at the weaker point

and the material shows necking effect as shown in figure.

Figure 7-2: Necking Phenomenon for mild steel

7.4 Procedure

Measure the length of the specimen by a Vernier Caliper Fix the specimen between

jaws of UTM

Apply tensile load on the specimen by the hydraulic system

Increase the load gradually up to the point at which the specimen deforms Record the

load and extension or compression of the specimen


Length of specimen = 120mm

Length of the holding section = 50mm Area of the specimen= πr2 = 33.2mm

2

Diameter of specimen = 6.50mm Diameter of the holding section = 9.30mm

48

Table 7-1: Calculations for Mild Steel

Serial Specimen Diameter of the Breaking Load Breaking Strength

# Material specimen (mm) (kN) (Pa)

1 Plain Steel 6.50 21.1 630

7.6 Graph

7.7 Comments

Due to poor finishing of the specimen, the results and graph obtained were not accurate.

49

8 Lab Session – 8

8.1 Objective

To perform tensile test on a specimen of aluminum (Al) by using universal testing machine

(UTM).

8.2 Apparatus:

Specimen (Mild Steel)



8.3 Theory





because of the versatility of its application. The following tests can be performed with it: -

Tension Test

Compression Test

Bending Test

Figure 8-1: Universal Testing Machine for Aluminum

50



1. The loading unit

2. The control panels.

Loading Unit




UTM consists of:

1. Upper cross head

2. Lower cross head



Lower Cross Head


Table


Elongation Scale



Control Unit


indicating devices.





Hydraulic Control Devices:


These are:



51





and printing.

Extensometer


Necking







Figure 8-2: Necking Phenomenon for aluminum

Procedure:


jaws of UTM




8.4 Observations and Calculations:


Length of the holding section = 50.0mm Area of the specimen= πr2 = 44.2mm

2


52

Table 8-1: Calculations for Aluminium


# Material specimen (mm) (kN) (MPa)

1 Mild Steel 7.50 16.1 364

8.5 Graph

8.6 Comments

Due to some error in the UTM, the results were not accurate as expected. Moreover, the

graph obtained is not according the expected results.

53

9 Lab Session – 9

9.1 Objective

To perform tensile test on a specimen of copper (Cu) by using universal testing machine

(UTM).

9.2 Apparatus

Specimen (Aluminum)



9.3 Theory






Tension Test

Compression Test

Bending Test

Figure 9-1: Universal Testing Machine for copper

54



1. The loading unit


Loading Unit




UTM consists of:

1. Upper cross head

2. Lower cross head



Lower Cross Head


Table


Elongation Scale



Control Unit


indicating devices.







These are:



55





and printing.

Extensometer


Necking







Figure 9-2: Necking Phenomenon for copper

9.4 Procedure


jaws of UTM





Length of specimen = 68.0mm

Length of the holding section = 50.0mm Area of the specimen= πr2 = 28.3mm

2


56

Table 9-1: Calculations for Copper



1 Aluminum 6.00 24.9 880

9.6 Graph

9.7 Comments

The finale results obtained were close to the theorized results. The graph shows minimal

distortion

57

10 Lab Session – 10

10.1 Objective

To perform tensile test on a specimen of brass by using universal testing machine (UTM).

10.2 Apparatus

Specimen (Brass)



10.3 Theory






Tension Test

Compression Test

Bending Test

Figure 10-1: Universal Testing Machine for Brass



58

3. The loading unit


Loading Unit




UTM consists of:

4. Upper cross head

5. Lower cross head



Lower Cross Head


Table


Elongation Scale



Control Unit


indicating devices.







These are:




59




and printing.

Extensometer


Necking







Figure 10-2: Necking Phenomenon for Brass

10.4 Procedure


jaws of UTM






Length of the holding section = 50mm Area of the specimen= πr2 = 38.5mm

2


60

Table 10-1: Calculations for Brass


# Material specimen (mm) (kN) (Pa)

1 Brass 7.00 11.2 290

10.6 Graph

10.7 Comments

Due to poor finishing of the specimen, the results and graph obtained were not accurate

61


11.1 Objective

To perform tensile test on a specimen of plain steel alloy by using universal testing machine

(UTM).

11.2 Apparatus

Specimen (Copper)



11.3 Theory





because of the versatility of its application. The following tests can be performed with it:

Tension Test

Compression Test

Bending Test

Figure 11-1: Universal Testing Machine for plain steel alloy

62



1. The loading unit


Loading Unit




UTM consists of:

1. Upper cross head

2. Lower cross head



Lower Cross Head


Table


Elongation Scale



Control Unit


indicating devices.







These are:



63





and printing.

Extensometer


Necking







Figure 11-2: Necking Phenomenon for plain steel alloy

11.4 Procedure


jaws of UTM





Length of specimen = 95.0mm

Length of the holding section = 50mm

Area of the specimen= πr2 = 28.3mm

2


64

Table 11-1: Calculations for Plain Steel Alloy



1 Copper 6.00 13.0 460

11.6 Graph

11.7 Comments

The machine error caused a few anomalies in the readings and eventually in the graph also.

65


12.1 Objective

To perform tensile test on a specimen of flat plate of polypropylene materials by using universal

testing machine (UTM).

12.2 Apparatus

Specimen (Polypropylene) Universal Testing Machine Control Unit


12.3 Theory






Tension Test

Compression Test

Bending Test

Figure 12-1: Universal Testing Machine for flat plate of polypropylene

66



1. The loading unit


Loading Unit




UTM consists of:

1. Upper cross head

2. Lower cross head



Lower Cross Head


Table


Elongation Scale



Control Unit


indicating devices.







These are:



67





and printing.

Extensometer


Necking



necking effect of a material, a tensile test is conducted on a specimen of the material. At a

certain maximum stress, the diameter of a portion of the specimen start to decrease at the

weaker point and the material shows necking effect as shown in figure.

Figure 12-2: Necking Phenomenon for flat plate of polypropylene

12.4 Procedure


jaws of UTM






Breadth of specimen = 10.0mm

Length of the holding section = 50mm

Height of specimen = 4.50mm

Area of the specimen= πr2 = 45.0mm

2

Table 12-1: Calculations for Flat Plate of Polypropylene

Serial Specimen Breadth of specimen Breaking Load Breaking Strength

# Material mm (kN) (Pa)

1 Polypropylene 10.0 3.40 3.40

68

12.6 Graphs

12.7 Comments

Due to poor finishing of the specimen, the results and graph obtained were not accurate. And

the exact value of breaking strength cannot be determined

69


13.1 Objective

To perform compression test on a specimen of concrete by using universal testing machine

(UTM).

13.2 Apparatus

Specimen (Concrete)



13.3 Theory






Tension Test

Compression Test

Bending Test

Figure 13-1: Universal Testing Machine for concrete

70



1. The loading unit


Loading Unit




UTM consists of:

1. Upper cross head

2. Lower cross head



Lower Cross Head


Table


Elongation Scale



Control Unit


indicating devices.







These are:



71





and printing.

Extensometer


Necking






Figure 13-2: Necking Phenomenon for concrete

13.4 Procedure


jaws of UTM





Area of the specimen= πr2 = 1140mm

2

Table 13-1: Calculations for Concrete

Serial Specimen Material

Diameter of specimen

(mm)

Breaking Load

(kN)

Breaking Strength

(Pa)

1 Concrete 10.0 1.48 1.30

72

13.6 Graphs

13.7 Comments

The strength of the material is not justifiable due to large area of the specimen.

73


14.1 Objective

To perform bending test on different beams of different materials by using universal testing

machine (UTM).

14.2 Apparatus:

Specimens



14.3 Theory






Tension Test

Compression Test

Bending Test

Figure 14-1: Universal Testing Machine for different beams

74



3. The loading unit


Loading Unit




UTM consists of:

4. Upper cross head

5. Lower cross head



Lower Cross Head


Table


Elongation Scale



Control Unit


indicating devices.







These are:



75





and printing.

Extensometer


Necking






Figure 14-2: Necking Phenomenon for different beams

14.4 Procedure


jaws of UTM




14.5 Observations and Calculations:

Table 14-1: Calculations for different beams

Sr.

No.

Specimen

Material

Length of

Specimen

(mm)

Area of the

specimen

(mm2)

Diameter or

Breadth of

specimen

(mm)

Breaking

Load

(kN)

Flexural

Strength

(Pa)

1 Mild Steel 205 50.0 10.0 1.00 1830

76

14.6 Graphs

14.7 Comments

The strength of the material is not justifiable due to large area of the specimen.

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